Solve 0.72=36/144+2x+x^2*x

$$0.72= \frac{ 36 }{ 144+2x+ { x }^{ 2 } } \times x$$

Solve for x

$$x=12\sqrt{3}+24\approx 44.784609691$$

$$x=24-12\sqrt{3}\approx 3.215390309$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by x^{2}+2x+144.

$$0.72\left(x^{2}+2x+144\right)=36x$$

Use the distributive property to multiply 0.72 by x^{2}+2x+144.

$$0.72x^{2}+1.44x+103.68=36x$$

Subtract 36x from both sides.

$$0.72x^{2}+1.44x+103.68-36x=0$$

Combine 1.44x and -36x to get -34.56x.

$$0.72x^{2}-34.56x+103.68=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.72 for a, -34.56 for b, and 103.68 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-34.56\right)±\sqrt{\left(-34.56\right)^{2}-4\times 0.72\times 103.68}}{2\times 0.72}$$

Square -34.56 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-34.56\right)±\sqrt{1194.3936-4\times 0.72\times 103.68}}{2\times 0.72}$$

Multiply -4 times 0.72.

$$x=\frac{-\left(-34.56\right)±\sqrt{1194.3936-2.88\times 103.68}}{2\times 0.72}$$

Multiply -2.88 times 103.68 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-34.56\right)±\sqrt{\frac{746496-186624}{625}}}{2\times 0.72}$$

Add 1194.3936 to -298.5984 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-34.56\right)±\sqrt{895.7952}}{2\times 0.72}$$

Take the square root of 895.7952.

$$x=\frac{-\left(-34.56\right)±\frac{432\sqrt{3}}{25}}{2\times 0.72}$$

The opposite of -34.56 is 34.56.

$$x=\frac{34.56±\frac{432\sqrt{3}}{25}}{2\times 0.72}$$

Multiply 2 times 0.72.

$$x=\frac{34.56±\frac{432\sqrt{3}}{25}}{1.44}$$

Now solve the equation x=\frac{34.56±\frac{432\sqrt{3}}{25}}{1.44} when ± is plus. Add 34.56 to \frac{432\sqrt{3}}{25}.

$$x=\frac{432\sqrt{3}+864}{1.44\times 25}$$

Divide \frac{864+432\sqrt{3}}{25} by 1.44 by multiplying \frac{864+432\sqrt{3}}{25} by the reciprocal of 1.44.

$$x=12\sqrt{3}+24$$

Now solve the equation x=\frac{34.56±\frac{432\sqrt{3}}{25}}{1.44} when ± is minus. Subtract \frac{432\sqrt{3}}{25} from 34.56.

$$x=\frac{864-432\sqrt{3}}{1.44\times 25}$$

Divide \frac{864-432\sqrt{3}}{25} by 1.44 by multiplying \frac{864-432\sqrt{3}}{25} by the reciprocal of 1.44.

$$x=24-12\sqrt{3}$$

The equation is now solved.

$$x=12\sqrt{3}+24 x=24-12\sqrt{3}$$

Steps for Completing the Square

Multiply both sides of the equation by x^{2}+2x+144.

$$0.72\left(x^{2}+2x+144\right)=36x$$

Use the distributive property to multiply 0.72 by x^{2}+2x+144.

$$0.72x^{2}+1.44x+103.68=36x$$

Subtract 36x from both sides.

$$0.72x^{2}+1.44x+103.68-36x=0$$

Combine 1.44x and -36x to get -34.56x.

$$0.72x^{2}-34.56x+103.68=0$$

Subtract 103.68 from both sides. Anything subtracted from zero gives its negation.

$$0.72x^{2}-34.56x=-103.68$$

Divide both sides of the equation by 0.72, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.72x^{2}-34.56x}{0.72}=-\frac{103.68}{0.72}$$

Dividing by 0.72 undoes the multiplication by 0.72.

$$x^{2}+\left(-\frac{34.56}{0.72}\right)x=-\frac{103.68}{0.72}$$

Divide -34.56 by 0.72 by multiplying -34.56 by the reciprocal of 0.72.

$$x^{2}-48x=-\frac{103.68}{0.72}$$

Divide -103.68 by 0.72 by multiplying -103.68 by the reciprocal of 0.72.

$$x^{2}-48x=-144$$

Divide -48, the coefficient of the x term, by 2 to get -24. Then add the square of -24 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-48x+\left(-24\right)^{2}=-144+\left(-24\right)^{2}$$

Square -24.

$$x^{2}-48x+576=-144+576$$

Add -144 to 576.

$$x^{2}-48x+576=432$$

Factor x^{2}-48x+576. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-24\right)^{2}=432$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-24\right)^{2}}=\sqrt{432}$$

Simplify.

$$x-24=12\sqrt{3} x-24=-12\sqrt{3}$$

Add 24 to both sides of the equation.

$$x=12\sqrt{3}+24 x=24-12\sqrt{3}$$